Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.7% → 98.9%

Time: 3.8s

Alternatives: 12

Speedup: 1.1×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (fma (* 0.0625 t) z (fma y x (* -0.25 (* b a)))) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma((0.0625 * t), z, fma(y, x, (-0.25 * (b * a)))) + c;
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(fma(Float64(0.0625 * t), z, fma(y, x, Float64(-0.25 * Float64(b * a)))) + c)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

   3. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

   5. lift-/.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

   6. mult-flip N/A

      \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

   11. lift-/.f64 N/A

       \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

   12. mult-flip N/A

       \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

   14. *-commutative N/A

       \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

   15. lower--.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

   16. associate--l+ N/A

       \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

   17. associate-*r* N/A

       \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

   18. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

   20. fp-cancel-sub-sign-inv N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

   21. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

   23. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

3. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]
4. Add Preprocessing

Alternative 2: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0.25 \cdot \left(b \cdot a\right)\\ t_2 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right) - t\_1\\ \mathbf{elif}\;t\_2 \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, x \cdot y\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* 0.25 (* b a))) (t_2 (/ (* a b) 4.0)))
   (if (<= t_2 -5e+48)
     (- (fma (* t z) 0.0625 c) t_1)
     (if (<= t_2 50000000000.0)
       (+ (fma (* 0.0625 t) z (* x y)) c)
       (- (fma y x c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = 0.25 * (b * a);
	double t_2 = (a * b) / 4.0;
	double tmp;
	if (t_2 <= -5e+48) {
		tmp = fma((t * z), 0.0625, c) - t_1;
	} else if (t_2 <= 50000000000.0) {
		tmp = fma((0.0625 * t), z, (x * y)) + c;
	} else {
		tmp = fma(y, x, c) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(0.25 * Float64(b * a))
	t_2 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_2 <= -5e+48)
		tmp = Float64(fma(Float64(t * z), 0.0625, c) - t_1);
	elseif (t_2 <= 50000000000.0)
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(x * y)) + c);
	else
		tmp = Float64(fma(y, x, c) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+48], N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 50000000000.0], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(y * x + c), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.99999999999999973e48

   1. Initial program 95.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + \frac{1}{16} \cdot \left(t \cdot z\right)\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(t \cdot z\right) \cdot \frac{1}{16} + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      8. lower-*.f64 82.9

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   if -4.99999999999999973e48 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5e10

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y}\right) + c \]
   5. Step-by-step derivation

      1. lower-*.f64 95.8

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, x \cdot \color{blue}{y}\right) + c \]

   6. Applied rewrites 95.8%

      \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \color{blue}{x \cdot y}\right) + c \]

   if 5e10 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 80.0

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, x \cdot y\right) + c\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (+ (fma (* 0.0625 t) z (* x y)) c)))
   (if (<= t_1 -1e+21)
     t_2
     (if (<= t_1 4e+195) (- (fma y x c) (* 0.25 (* b a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((0.0625 * t), z, (x * y)) + c;
	double tmp;
	if (t_1 <= -1e+21) {
		tmp = t_2;
	} else if (t_1 <= 4e+195) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(fma(Float64(0.0625 * t), z, Float64(x * y)) + c)
	tmp = 0.0
	if (t_1 <= -1e+21)
		tmp = t_2;
	elseif (t_1 <= 4e+195)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+21], t$95$2, If[LessEqual[t$95$1, 4e+195], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1e21 or 3.99999999999999991e195 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y}\right) + c \]
   5. Step-by-step derivation

      1. lower-*.f64 85.2

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, x \cdot \color{blue}{y}\right) + c \]

   6. Applied rewrites 85.2%

      \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, \color{blue}{x \cdot y}\right) + c \]

   if -1e21 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 3.99999999999999991e195

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 90.9

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+195}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (+ (fma (* t z) 0.0625 (* y x)) c)))
   (if (<= t_1 -1e+21)
     t_2
     (if (<= t_1 4e+195) (- (fma y x c) (* 0.25 (* b a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((t * z), 0.0625, (y * x)) + c;
	double tmp;
	if (t_1 <= -1e+21) {
		tmp = t_2;
	} else if (t_1 <= 4e+195) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(fma(Float64(t * z), 0.0625, Float64(y * x)) + c)
	tmp = 0.0
	if (t_1 <= -1e+21)
		tmp = t_2;
	elseif (t_1 <= 4e+195)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(y * x), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+21], t$95$2, If[LessEqual[t$95$1, 4e+195], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1e21 or 3.99999999999999991e195 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} + c \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(t \cdot z\right) \cdot \frac{1}{16} + \color{blue}{x} \cdot y\right) + c \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{\frac{1}{16}}, x \cdot y\right) + c \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, x \cdot y\right) + c \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, y \cdot x\right) + c \]

      5. lower-*.f64 83.7

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c \]

   4. Applied rewrites 83.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)} + c \]

   if -1e21 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 3.99999999999999991e195

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 90.9

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+170}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* t z) 0.0625 c)))
   (if (<= t_1 -2e+170)
     t_2
     (if (<= t_1 2e+208) (- (fma y x c) (* 0.25 (* b a))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((t * z), 0.0625, c);
	double tmp;
	if (t_1 <= -2e+170) {
		tmp = t_2;
	} else if (t_1 <= 2e+208) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (t_1 <= -2e+170)
		tmp = t_2;
	elseif (t_1 <= 2e+208)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+170], t$95$2, If[LessEqual[t$95$1, 2e+208], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2.00000000000000007e170 or 2e208 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c + \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, 0.0625, c\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} + c \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) \]

      4. lower-*.f64 81.2

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) \]

   9. Applied rewrites 81.2%

      \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right) \]

   if -2.00000000000000007e170 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e208

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 88.3

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;c - 0.25 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -2e+95)
     (- c (* 0.25 (* a b)))
     (if (<= t_1 4e-209)
       (fma (* t z) 0.0625 c)
       (if (<= t_1 5e+137) (fma y x c) (fma -0.25 (* b a) (* y x)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = c - (0.25 * (a * b));
	} else if (t_1 <= 4e-209) {
		tmp = fma((t * z), 0.0625, c);
	} else if (t_1 <= 5e+137) {
		tmp = fma(y, x, c);
	} else {
		tmp = fma(-0.25, (b * a), (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = Float64(c - Float64(0.25 * Float64(a * b)));
	elseif (t_1 <= 4e-209)
		tmp = fma(Float64(t * z), 0.0625, c);
	elseif (t_1 <= 5e+137)
		tmp = fma(y, x, c);
	else
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], N[(c - N[(0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-209], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[t$95$1, 5e+137], N[(y * x + c), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000004e95

   1. Initial program 95.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c + \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, 0.0625, c\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto c - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.3%

      \[\leadsto c - \color{blue}{0.25} \cdot \left(a \cdot b\right) \]

   if -2.00000000000000004e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.0000000000000002e-209

   1. Initial program 99.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c + \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, 0.0625, c\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} + c \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) \]

      4. lower-*.f64 62.1

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) \]

   9. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right) \]

   if 4.0000000000000002e-209 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.0000000000000002e137

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 69.9

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. lift-fma.f64 57.8

         \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   7. Applied rewrites 57.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]

   if 5.0000000000000002e137 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 84.9

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in c around 0

      \[\leadsto x \cdot y - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot y + \frac{-1}{4} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(a \cdot b\right) + x \cdot \color{blue}{y} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, a \cdot \color{blue}{b}, x \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, x \cdot y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, x \cdot y\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4}, b \cdot a, y \cdot x\right) \]

      8. lift-*.f64 79.1

         \[\leadsto \mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right) \]

   7. Applied rewrites 79.1%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{b \cdot a}, y \cdot x\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 64.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := c - 0.25 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (- c (* 0.25 (* a b)))))
   (if (<= t_1 -2e+95)
     t_2
     (if (<= t_1 4e-209)
       (fma (* t z) 0.0625 c)
       (if (<= t_1 1e+42) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = c - (0.25 * (a * b));
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = t_2;
	} else if (t_1 <= 4e-209) {
		tmp = fma((t * z), 0.0625, c);
	} else if (t_1 <= 1e+42) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(c - Float64(0.25 * Float64(a * b)))
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = t_2;
	elseif (t_1 <= 4e-209)
		tmp = fma(Float64(t * z), 0.0625, c);
	elseif (t_1 <= 1e+42)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(c - N[(0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], t$95$2, If[LessEqual[t$95$1, 4e-209], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[t$95$1, 1e+42], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000004e95 or 1.00000000000000004e42 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c + \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, 0.0625, c\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   7. Taylor expanded in z around 0

      \[\leadsto c - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 68.6%

      \[\leadsto c - \color{blue}{0.25} \cdot \left(a \cdot b\right) \]

   if -2.00000000000000004e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.0000000000000002e-209

   1. Initial program 99.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c + \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, 0.0625, c\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} + c \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) \]

      4. lower-*.f64 62.1

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) \]

   9. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right) \]

   if 4.0000000000000002e-209 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.00000000000000004e42

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 68.6

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. lift-fma.f64 61.9

         \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -2e+95)
     t_2
     (if (<= t_1 4e-209)
       (fma (* t z) 0.0625 c)
       (if (<= t_1 2e+235) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -2e+95) {
		tmp = t_2;
	} else if (t_1 <= 4e-209) {
		tmp = fma((t * z), 0.0625, c);
	} else if (t_1 <= 2e+235) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -2e+95)
		tmp = t_2;
	elseif (t_1 <= 4e-209)
		tmp = fma(Float64(t * z), 0.0625, c);
	elseif (t_1 <= 2e+235)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+95], t$95$2, If[LessEqual[t$95$1, 4e-209], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+235], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000004e95 or 2.0000000000000001e235 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 93.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 70.2

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 70.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -2.00000000000000004e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.0000000000000002e-209

   1. Initial program 99.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{x \cdot y} + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \frac{a \cdot b}{4}\right) + c \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \frac{a \cdot b}{4}\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{z \cdot t}}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{z \cdot t}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      6. mult-flip N/A

         \[\leadsto \left(\left(\color{blue}{\left(z \cdot t\right) \cdot \frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(t \cdot z\right)} \cdot \frac{1}{16} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + x \cdot y\right) - \frac{a \cdot b}{4}\right) + c \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{\color{blue}{a \cdot b}}{4}\right) + c \]

      11. lift-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{a \cdot b}{4}}\right) + c \]

      12. mult-flip N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\left(a \cdot b\right) \cdot \frac{1}{4}}\right) + c \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \left(a \cdot b\right) \cdot \color{blue}{\frac{1}{4}}\right) + c \]

      14. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)}\right) + c \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      16. associate--l+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right)} + c \]

      17. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{16} \cdot t\right) \cdot z} + \left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)\right) + c \]

      18. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{16} \cdot t, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{16} \cdot t}, z, x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      20. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{x \cdot y + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{y \cdot x} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)\right) + c \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, y \cdot x + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right)\right) + c \]

      23. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right)}\right) + c \]

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\right)} + c \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c + \left(\frac{-1}{4} \cdot \left(a \cdot b\right) + \frac{1}{16} \cdot \left(t \cdot z\right)\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot t, 0.0625, c\right) - 0.25 \cdot \left(a \cdot b\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \left(t \cdot z\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} + c \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, c\right) \]

      4. lower-*.f64 62.1

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, c\right) \]

   9. Applied rewrites 62.1%

      \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{0.0625}, c\right) \]

   if 4.0000000000000002e-209 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.0000000000000001e235

   1. Initial program 99.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 71.3

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. lift-fma.f64 53.6

         \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   7. Applied rewrites 53.6%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+80}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -5e+80) t_2 (if (<= t_1 2e+235) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -5e+80) {
		tmp = t_2;
	} else if (t_1 <= 2e+235) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -5e+80)
		tmp = t_2;
	elseif (t_1 <= 2e+235)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+80], t$95$2, If[LessEqual[t$95$1, 2e+235], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.99999999999999961e80 or 2.0000000000000001e235 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 68.6

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 68.6%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -4.99999999999999961e80 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.0000000000000001e235

   1. Initial program 99.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      7. lower-*.f64 69.5

         \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   4. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot y + c \]

      2. *-commutative N/A

         \[\leadsto y \cdot x + c \]

      3. lift-fma.f64 59.8

         \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   7. Applied rewrites 59.8%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, c\right) \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 (fma y x c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma(y, x, c);
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(y, x, c)
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(y * x + c), $MachinePrecision]

Derivation

1. Initial program 97.7%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(c + x \cdot y\right) - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot y + c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

   3. *-commutative N/A

      \[\leadsto \left(y \cdot x + c\right) - \frac{1}{4} \cdot \left(a \cdot b\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{1}{4}} \cdot \left(a \cdot b\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - \frac{1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

   7. lower-*.f64 73.6

      \[\leadsto \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

4. Applied rewrites 73.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)} \]

5. Taylor expanded in a around 0

   \[\leadsto c + \color{blue}{x \cdot y} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot y + c \]

   2. *-commutative N/A

      \[\leadsto y \cdot x + c \]

   3. lift-fma.f64 48.8

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

7. Applied rewrites 48.8%

   \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, c\right) \]
8. Add Preprocessing

Alternative 11: 42.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+104}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+49}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5e+104) (* y x) (if (<= (* x y) 4e+49) c (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+104) {
		tmp = y * x;
	} else if ((x * y) <= 4e+49) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x * y) <= (-5d+104)) then
        tmp = y * x
    else if ((x * y) <= 4d+49) then
        tmp = c
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+104) {
		tmp = y * x;
	} else if ((x * y) <= 4e+49) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -5e+104:
		tmp = y * x
	elif (x * y) <= 4e+49:
		tmp = c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -5e+104)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 4e+49)
		tmp = c;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -5e+104)
		tmp = y * x;
	elseif ((x * y) <= 4e+49)
		tmp = c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+104], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+49], c, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.9999999999999997e104 or 3.99999999999999979e49 < (*.f64 x y)

   1. Initial program 95.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{x} \]

      2. lower-*.f64 62.5

         \[\leadsto y \cdot \color{blue}{x} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{y \cdot x} \]

   if -4.9999999999999997e104 < (*.f64 x y) < 3.99999999999999979e49

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c} \]
   3. Step-by-step derivation

   4. Applied rewrites 29.8%

      \[\leadsto \color{blue}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 22.5% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ c \end{array} \]

FPCore

(FPCore (x y z t a b c) :precision binary64 c)

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return c;
}

Python

def code(x, y, z, t, a, b, c):
	return c

Julia

function code(x, y, z, t, a, b, c)
	return c
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := c

Derivation

1. Initial program 97.7%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{c} \]
3. Step-by-step derivation

4. Applied rewrites 22.5%

   \[\leadsto \color{blue}{c} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))